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Theorem sepnsepolem2 47029
Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 47030. Proof could be shortened by 1 step using ssdisjdr 46967. (Contributed by Zhi Wang, 1-Sep-2024.)
Hypothesis
Ref Expression
sepnsepolem2.1 (πœ‘ β†’ 𝐽 ∈ Top)
Assertion
Ref Expression
sepnsepolem2 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝐷(π‘₯)   𝐽(π‘₯)

Proof of Theorem sepnsepolem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sepnsepolem2.1 . 2 (πœ‘ β†’ 𝐽 ∈ Top)
2 id 22 . . 3 (𝐽 ∈ Top β†’ 𝐽 ∈ Top)
3 sslin 4199 . . . . 5 (𝑧 βŠ† 𝑦 β†’ (π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦))
4 sseq0 4364 . . . . . 6 (((π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦) ∧ (π‘₯ ∩ 𝑦) = βˆ…) β†’ (π‘₯ ∩ 𝑧) = βˆ…)
54ex 414 . . . . 5 ((π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
63, 5syl 17 . . . 4 (𝑧 βŠ† 𝑦 β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
76adantl 483 . . 3 ((𝐽 ∈ Top ∧ 𝑧 βŠ† 𝑦) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
8 ineq2 4171 . . . . 5 (𝑦 = 𝑧 β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ 𝑧))
98eqeq1d 2739 . . . 4 (𝑦 = 𝑧 β†’ ((π‘₯ ∩ 𝑦) = βˆ… ↔ (π‘₯ ∩ 𝑧) = βˆ…))
109adantl 483 . . 3 ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) β†’ ((π‘₯ ∩ 𝑦) = βˆ… ↔ (π‘₯ ∩ 𝑧) = βˆ…))
112, 7, 10opnneieqv 47017 . 2 (𝐽 ∈ Top β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
121, 11syl 17 1 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  β€˜cfv 6501  Topctop 22258  neicnei 22464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-top 22259  df-nei 22465
This theorem is referenced by:  sepnsepo  47030
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